Editing Wilson loop (section)

==Additional applications==

===Scattering amplitudes===

Wilson loops play a role in the theory of [[scattering amplitude]]s where a set of dualities between them and special types of scattering amplitudes has been found.<ref>{{cite journal|last1=Alday|first1=L.F.|authorlink1=Luis Fernando Alday|last2=Radu|first2=R.|authorlink2=|date=2008|title=Scattering Amplitudes, Wilson Loops and the String/Gauge Theory Correspondence|url=|journal=Phys. Rep.|volume=468|issue=5|pages=153–211|doi=10.1016/j.physrep.2008.08.002|pmid=|arxiv=0807.1889|bibcode=2008PhR...468..153A |s2cid=119220578|access-date=}}</ref> These have first been suggested at strong coupling using the [[AdS/CFT correspondence]].<ref>{{cite journal|last1=Alday|first1=L.F.|authorlink1=Luis Fernando Alday|last2=Maldacena|first2=J.M.|authorlink2=Juan Martín Maldacena|date=2007|title=Gluon scattering amplitudes at strong coupling|url=|journal=JHEP|volume=6|issue=6|page=64|doi=10.1088/1126-6708/2007/06/064|pmid=|arxiv=0705.0303|bibcode=2007JHEP...06..064A |s2cid=10711473|access-date=}}</ref> For example, in <math>\mathcal N=4</math> [[N = 4 supersymmetric Yang–Mills theory|supersymmetric Yang–Mills theory]] [[MHV amplitudes|maximally helicity violating amplitudes]] factorize into a tree-level component and a loop level correction.<ref>{{cite book|last=Henn|first=J.M.|author-link=:de:Johannes Henn|date=2014|title=Scattering Amplitudes in Gauge Theories|url=|doi=|location=|publisher=Springer|chapter=4|pages=153–158|isbn=978-3642540219}}</ref> This loop level correction does not depend on the [[helicity (particle physics)|helicities]] of the particles, but it was found to be dual to certain polygonal Wilson loops in the large <math>N</math> limit, up to finite terms. While this duality was initially only suggested in the maximum helicity violating case, there are arguments that it can be extended to all helicity configurations by defining appropriate [[supersymmetry|supersymmetric]] generalizations of the Wilson loop.<ref>{{cite journal|last1=Caron-Huot|first1=S.|authorlink1=:de:Simon Caron-Huot|date=2011|title=Notes on the scattering amplitudes/Wilson loop duality|url=|journal=JHEP|volume=2011|issue=7|page=58|doi=10.1007/JHEP07(2011)058|pmid=|arxiv=1010.1167|bibcode=2011JHEP...07..058C |s2cid=118676335|access-date=}}</ref>

===String theory compactifications===

In [[compactification (physics)|compactified]] theories, zero mode gauge field states that are locally pure gauge configurations but are globally inequivalent to the vacuum are parameterized by closed Wilson lines in the compact direction. The presence of these on a compactified [[String (physics)|open]] [[string theory]] is equivalent under [[T-duality]] to a theory with non-coincident [[D-brane|D-branes]], whose separations are determined by the Wilson lines.<ref>{{cite book|last=Polchinski|first=J.|author-link=Joseph Polchinski|date=1998|title=String Theory Volume I: An Introduction to the Bosonic String|url=|doi=|location=|publisher=Cambridge University Press|chapter=8|pages=263–268|isbn=978-0143113799}}</ref> Wilson lines also play a role in [[orbifold]] compactifications where their presence leads to greater control of gauge [[symmetry breaking]], giving a better handle on the final unbroken gauge group and also providing a mechanism for controlling the number of matter multiplets left after compactification.<ref>{{cite journal|last1=Ibanez|first1=L.E.|authorlink1=|last2=Nilles|first2=H.P.|authorlink2=|last3=Quevedo|first3=F.|authorlink3=Fernando Quevedo|date=1986|title=Orbifolds and Wilson Lines|url=https://cds.cern.ch/record/173902|journal=Phys. Lett. B|volume=187|issue=1–2|pages=25–32|doi=10.1016/0370-2693(87)90066-9|pmid=|arxiv=|s2cid=|access-date=}}</ref> These properties make Wilson lines important in compactifications of superstring theories.<ref>{{cite book|last=Polchinski|first=J.|author-link=Joseph Polchinski|date=1998|title=String Theory Volume II: Superstring Theory and Beyond|url=|doi=|location=|publisher=Cambridge University Press|chapter=16|pages=288–290|isbn=978-1551439761}}</ref><ref>{{cite book|last1=Choi|first1=K.S.|author-link1=|last2=Kim|first2=J.E.|author-link2=|date=2020|title=Quarks and Leptons From Orbifolded Superstring|edition=2|url=|doi=|location=|publisher=|chapter=|page=|isbn=978-3030540043}}</ref>

===Topological field theory===

In a [[topological quantum field theory|topological field theory]], the expectation value of Wilson loops does not change under smooth deformations of the loop since the field theory does not depend on the [[metric (mathematics)|metric]].<ref>{{cite book|last=Fradkin|first=E.|author-link=Eduardo Fradkin|date=2021|title=Quantum Field Theory: An Integrated Approach|url=|doi=|location=|publisher=Princeton University Press|chapter=22|page=697|isbn=978-0691149080}}</ref> For this reason, Wilson loops are key [[observable]]s on in these theories and are used to calculate global properties of the [[manifold]]. In <math>2+1</math> dimensions they are closely related to [[knot theory]] with the expectation value of a product of loops depending only on the manifold structure and on how the loops are tied together. This led to the famous connection made by [[Edward Witten]] where he used Wilson loops in [[Chern–Simons theory]] to relate their [[partition function (quantum field theory)|partition function]] to [[Jones polynomial]]s of knot theory.<ref>{{cite journal|last1=Witten|first1=E.|authorlink1=Edward Witten|date=1989|title=Quantum Field Theory and the Jones Polynomial|url=http://projecteuclid.org/euclid.cmp/1104178138|journal=Commun. Math. Phys.|volume=121|issue=3|pages=351–399|doi=10.1007/BF01217730|pmid=|arxiv=|bibcode=1989CMaPh.121..351W |s2cid=14951363|access-date=}}</ref>